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Characterizing and quantifying quantum entanglement in states of open quantum systems are at the heart of quantum information theory. In this work, we study the mixedness and entanglement of a two-mode symmetrical Gaussian state evolving in a thermal Markovian environment. The considered state is that of two movable end-mirrors of two distant Fabry–Pérot cavities coupled via a two-mode squeezed light and driven by coherent lasers. We use the logarithmic negativity to consistently quantify the degree of entanglement between the two mechanical modes. The purity is employed as a witness of mixedness and characterize of entanglement in the two-mode mechanical bi-partition. By comparing the behavior of the logarithmic negativity with that of the purity under the effect of various parameters and based on the entanglement classification established in [G. Adesso, A. Serafini and F. Illuminati, Phys. Rev. Lett.92, 087901 (2004)], we found that the purity can be used as a characterize and measure of entanglement in the two-mode symmetrical state when the squeezing parameter is fixed. The effects of thermal noise, squeezing and optomechanical coupling are also discussed in detail. [ABSTRACT FROM AUTHOR]

The Dzyaloshinsky–Moriya (DM) interaction contributes to some unusual and interesting magnetic properties in real materials, thus playing an important role in the degree of quantum entanglement in Heisenberg quantum spin models. In [C. S. Castro, O. S. Duarte, D. P. Pires, D. O. Soares-Pinto and M. S. Reis, Phys. Lett. A380, 1571 (2016)], it has been investigated about the non-locality and the thermal entanglement in a dipolar spin thermal system without DM interaction. In this work, we study the entanglement in the thermal state of inhomogeneous Heisenberg coupling under the presence of the DM interaction along the z -axis. More precisely, we analyze the effect of the DM interaction on non-locality phenomena and quantum entanglement as measured by negativity and Von Neumann entropy. We show that by comparing with [C. S. Castro, O. S. Duarte, D. P. Pires, D. O. Soares-Pinto and M. S. Reis, Phys. Lett. A380, 1571 (2016)], the local quantum states become more pronounced, when the DM interaction is taken into account. This fact is well confirmed by noting that Von Neumann entropy is destroyed in the presence of the DM interaction. It can be deduced that the Dzyaloshinsky–Moriya (DM) interaction makes the thermal states less correlated. [ABSTRACT FROM AUTHOR]